A scatter plot is a graph that shows how two quantitative variables are related by plotting paired data values as points. The x-axis usually shows the explanatory variable, and the y-axis usually shows the response variable. Scatter plots matter because they help you see patterns that are hard to notice in a table of numbers.
They are used in science, economics, sports, health, and engineering to study relationships between measurements.
To read a scatter plot, look at the direction, form, and strength of the point pattern. Direction tells whether y tends to increase or decrease as x increases, form tells whether the pattern is roughly linear or curved, and strength tells how closely the points follow the pattern. Outliers and clusters are important because they may reveal unusual cases, measurement errors, or hidden groups in the data.
Correlation measures the strength and direction of a linear relationship, but it does not prove that one variable causes the other.
Understanding Statistics: Scatter Plots
The scale of each axis can strongly affect what a pattern seems to show. A graph with a very wide scale can make a relationship look weak, while a narrow scale can make small differences look dramatic. Start the axes at sensible values and use equal intervals.
Labels need units, such as centimetres, minutes, kilograms, or degrees Celsius. It is important that each point comes from the same individual, object, or event. Pairing one student's height with another student's shoe size would create meaningless data.
A trend line is a model, not a path that every point must follow. Its job is to describe the centre of a roughly straight cloud of points. A useful way to judge the line is to examine residuals.
A residual is the vertical difference between an actual data value and the value predicted by the line. Points above the line have positive residuals. Points below it have negative residuals.
If residuals appear randomly scattered, a straight-line model may be reasonable. If they make a curve or spread out more at one end, the line misses an important feature of the data.
Trend lines can be used for interpolation, which means estimating a response for an explanatory value inside the observed data range. For example, a student might estimate fuel use for a trip distance between distances already measured. Predictions become much less reliable outside that range.
This is called extrapolation. A line based on plants growing from five to twenty centimetres tall cannot safely predict what happens at two metres. Conditions may change, physical limits may appear, and a curved pattern may no longer behave like a line.
Real data often contain influences that a graph does not display. Ice cream sales and sunburn cases may rise together because hot weather affects both. The sales do not necessarily cause the sunburn.
This hidden influence is called a confounding variable. Groups can create another trap. Data from younger and older students may form separate clusters, and combining them can hide the pattern within each age group.
When studying scatter plots, describe the visible evidence carefully. State the overall trend, note unusual points, consider possible explanations, and avoid claims that go beyond the data. Check how the data were collected before trusting a conclusion.
Key Facts
- Each point in a scatter plot represents one ordered pair: (x, y).
- The explanatory variable is usually placed on the x-axis, and the response variable is usually placed on the y-axis.
- A positive association means y tends to increase as x increases.
- A negative association means y tends to decrease as x increases.
- The correlation coefficient r satisfies -1 ≤ r ≤ 1 and describes the direction and strength of a linear relationship.
- A line of best fit can be written as y = mx + b, where m is the slope and b is the y-intercept.
Vocabulary
- Scatter plot
- A graph that displays paired numerical data as points on a coordinate plane.
- Explanatory variable
- The variable plotted on the x-axis that may help explain or predict changes in another variable.
- Response variable
- The variable plotted on the y-axis that is measured as an outcome.
- Correlation
- A measure of the direction and strength of a linear relationship between two quantitative variables.
- Outlier
- A data point that lies far away from the overall pattern of the other points.
Common Mistakes to Avoid
- Putting the variables on the wrong axes is wrong because the explanatory variable usually belongs on the x-axis and the response variable belongs on the y-axis.
- Assuming correlation proves causation is wrong because two variables can be related without one directly causing the other.
- Ignoring outliers is wrong because an unusual point can strongly affect the apparent trend, correlation, and line of best fit.
- Describing only direction and forgetting strength and form is wrong because a complete scatter plot description should include direction, form, strength, and unusual features.
Practice Questions
- 1 A student records study time and test score as (1, 62), (2, 70), (3, 75), (4, 83), and (5, 88). Describe the direction of the association and estimate the change in score for each additional hour studied.
- 2 For a line of best fit y = 4.5x + 32, predict y when x = 6. Then interpret the slope in the context of a scatter plot.
- 3 A scatter plot shows a strong positive association between ice cream sales and number of people at a beach. Explain why this does not prove that ice cream sales cause more people to go to the beach.